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§4.4  Flow near Solid Surfaces  by  Boundary-Layer  Theory  139


                                 1.0



                                 0.8                       /




                                 0.6


                                                                               X
                                                                          R
                                 0.4
                                                                         + 1.08x1o 5
                                                                         * 1.82x1o 5
                                                                         o3.64x1o  5
                                 0.2                                     • ^.46x1n  5
                                                                         Д .28x1o 5
                                                                           7
                                      /
                                    /
                                            1.0     2.0      3.0      4.0      5.0  5.64  6.0   7.0



                           Fig. 4.4-3.  Predicted and observed  velocity  profiles  for  tangential laminar flow along a
                           flat plate. The solid  line represents the solution  of  Eqs. 4.4-20 to 24, obtained by  Blasius
                            [see H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, 7th edition (1979),
                           p. 137].



                                                                                       4
                           analyses 8  indicate that Eq. 4.4-30 is accurate to within  3% for  Lv /v  >  10  and within  0.3%
                                                                                x
                            iorLv /v>  10. 6
                                x
                               The growth  of  the boundary  layer  with  increasing  x eventually  leads  to an  unstable
                            situation, and  turbulent  flow  sets  in. The transition  is  found  to begin  somewhere  in the
                                                                                     6
                                                                            5
                            range  of  local  Reynolds  number  of  Re v  = xv /v  >  3  X  10  to 3  X  10 , depending  on the
                                                                 x
                           uniformity  of  the  approaching  stream.  8  Upstream  of  the  transition  region  the  flow  re-
                           mains laminar, and downstream  it is turbulent.
       EXAMPLE   4.4-3      We  now  want  to treat the boundary-layer  problem  analogous  to Example  4.3-3, namely  the
                            flow near  a corner  (see  Fig. 4.3-4).  If a  >  1, the problem  may  also  be interpreted as  the flow
      Flow near a Corner    along a wedge  of included angle  /Зтг, with a  = 2/(2  -  /3). For this system  the external flow v e
                            is known  from  Eqs. 4.3-42 and 43, where we  found that

                                                           v e (x)  =                           (4.4-31)
                           This was  the expression  that was  found  to be valid  right  at the wall  (i.e., at у  = 0). Here, it
                            is  assumed  that  the boundary  layer  is  so  thin  that  using  the  wall  expression  from  ideal
                            flow  is adequate for  the outer limit  of  the boundary-layer  solution, at least  for  small  values
                            of*.




                               * Y. H. Kuo, /. Math.  Phys.,  32, 83-101  (1953); I. Imai, /. Aero. Sci., 24,155-156 (1957).
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